We will first establish the liminf-estimates. Namely, consider a sequence
\((\mathsf {P}^n,\mathsf {J}^{n,+},\mathsf {J}^{n,-})\in \mathsf {CE}_n\)  that converges to the curve \((\mathsf {P},\mathsf {J}^+,\mathsf {J}^-)\in \mathsf {CE}_{\infty }\) . In particular \(\mathsf {P}_t^n\rightarrow \mathsf {P}_t\)  for all \(t\in [0,T]\) , and hence by Theorem REF  on the \(\Gamma \) -convergence of \(\mathcal {F}_n\)  we immediately obtain
\(\liminf _{n\rightarrow \infty } \mathcal {F}_n(\mathsf {P}_t^n)\ge \mathcal {F}_{\infty }(\mathsf {P}_t), \qquad \mbox{ for all }t\in [0,T].\) 
Now suppose that
\(\limsup _{n\rightarrow \infty } \mathcal {F}_n(\mathsf {P}_0^n)<\infty , \qquad \limsup _{n\rightarrow \infty } \mathcal {I}_n(\mathsf {P}^n,\mathsf {J}^{n,+},\mathsf {J}^{n,-})<\infty .\) 
In particular we have the bounds
\(\limsup _{n\rightarrow \infty } \int _0^T \mathcal {R}_n(\mathsf {P}^n_t,\mathsf {J}^{n,+},\mathsf {J}^{n,-}) \, \mathrm {d}t < \infty , \quad \limsup _{n\rightarrow \infty } \int _0^T \mathcal {D}_n(\mathsf {P}^n_t) \, \mathrm {d}t < \infty .\) 
Due to the chain rule and the assumption on \(\mathcal {F}_n(\mathsf {P}^n_0)\) , we obtain
\(\limsup _{n\rightarrow \infty } \sup _{t\in [0,T]} \mathcal {F}_n(\mathsf {P}^n_t) < \infty .\) 
The latter guarantees, by Corollary REF , that we have the vague convergence
\(\begin{aligned}\lim _{n\rightarrow \infty } \vartheta ^{\pm }_{\mathsf {P}_t^n} = \vartheta ^{\pm }_{\mathsf {P}_t}, \qquad \lim _{n\rightarrow \infty } \mathsf {T}^{n,\pm }_{\#}\vartheta _{\mathsf {P}_t^n}^{\pm } =\vartheta ^{\pm }_{\mathsf {P}_t}.\end{aligned}\) 
Recall that from Lemma REF  and Remark REF  we have for each \(n\ge 1\) :
\(\begin{aligned}\mathcal {E}\mathrm {nt}\left(\mathsf {J}_t^{n,\pm }|\Theta _{\mathsf {P}}^{n,+}\right)&=\int _{\Gamma \times \mathcal {T}} \Upsilon \left(\frac{\mathrm {d}\mathsf {J}_t^{n,\pm }}{\mathrm {d}\Sigma },\frac{\mathrm {d}\vartheta _{\mathsf {P}_t}^{\pm }}{\mathrm {d}\Sigma },\frac{\mathrm {d}(\mathsf {T}^{n,\mp }_{\#}\vartheta _{\mathsf {P}_t}^{\mp })}{\mathrm {d}\Sigma }\right)\mathrm {d}\Sigma ,\\\mathcal {D}_{n}(\mathsf {P}_t)&=2H^2(\vartheta _{\mathsf {P}_t}^{\pm },\mathsf {T}^{n,\mp }_{\#}\vartheta _{\mathsf {P}_t}^{\mp }),\end{aligned}\) 
for any dominating measure \(\Sigma \) , and similarly, from Corollary REF  and Remark REF  that
\(\begin{aligned}\mathcal {E}\mathrm {nt}\left(\mathsf {J}_t^{\pm }|\Theta _{\mathsf {P}}^{+}\right)&=\int _{\Gamma \times \mathcal {T}} \Upsilon \left(\frac{\mathrm {d}\vartheta _{\mathsf {P}_t}^{\pm }}{\mathrm {d}\Sigma },\frac{\mathrm {d}\vartheta _{\mathsf {P}_t}^{\pm }}{\mathrm {d}\Sigma },\frac{\mathrm {d}\vartheta _{\mathsf {P}_t}^{\mp }}{\mathrm {d}\Sigma }\right)\mathrm {d}\Sigma ,\\\mathcal {D}_{\infty }(\mathsf {P}_t)&=2H^2(\vartheta _{\mathsf {P}_t}^{\pm },\vartheta _{\mathsf {P}_t}^{\mp }).\end{aligned}\) 
By the convexity and lower semi-continuity of \(\Upsilon \)  and \(H\)  we conclude by standard semi-continuity results (e.g. see [1]}) that for each \(t\in [0,T]\) ,
\(\liminf _{n\rightarrow \infty } \mathcal {R}_n(\mathsf {P}_t^n,\mathsf {J}^{n,+}_t,\mathsf {J}^{n,-}_t) \ge \mathcal {R}_n(\mathsf {P}_t,\mathsf {J}^{+}_t,\mathsf {J}^{-}_t), \qquad \liminf _{n\rightarrow \infty } \mathcal {D}_n(\mathsf {P}_t^n) \ge \mathcal {D}_n(\mathsf {P}_t),\) 
from which (REF ) directly follows after applying the Fatou lemma.
Next, we consider the question of compactness. As in the previous part, let us consider a sequence \((\mathsf {P}^n,\mathsf {J}^{n,+},\mathsf {J}^{n,-})\in \mathsf {CE}_n\)  with
\(\limsup _{n\rightarrow \infty } \mathcal {F}_n(\mathsf {P}_0^n)<\infty , \qquad \limsup _{n\rightarrow \infty } \mathcal {I}_n(\mathsf {P}^n,\mathsf {J}^{n,+},\mathsf {J}^{n,-})<\infty ,\) 
which imply that the estimates (REF ) and (REF ) still hold. The bound on the free energy ensures by Theorem REF  that \(\lbrace \mathsf {P}_{t}^n\rbrace _{t\in [0,T],n\ge 1}\)  is pre-compact. Moreover, due to the bound on the action \(\mathcal {R}_n\) , we have by the results of Corollary (REF ) and Lemma (REF ) that
\(\limsup _{n\rightarrow \infty } \int _0^T \tilde{\phi }\left(\frac{1}{3M} \int _{\Gamma \times \mathcal {T}} (1+\nu (X)^2)^{-1}\,\mathsf {J}^{n,\pm }_t(\mathrm {d}\nu ,\mathrm {d}x) \right) \, \mathrm {d}t < \infty ,\) 
\(\limsup _{n\rightarrow \infty } \int _0^T \tilde{\phi }\left(\frac{|\dot{\mathsf {P}}^n_t|_{W}}{12M}\right) \, \mathrm {d}t<\infty ,\) 
where \(|\dot{\mathsf {P}}^n_t|_W\)  is again the \(W\) -metric speed. From (REF ), we then conclude from the non-decreasing, convex and super-linear at infinity property of \(\tilde{\phi }\)  that, up to choosing a subsequence \(n^{\prime }\) , there exists a family \(\lbrace \mathsf {J}^{\pm }_t\rbrace _{t\in [0,T]} \in \mathcal {M}_{loc}^+(\Gamma \times \mathcal {T})\)  such that for all \(s,t\)  the sequence of measures \(\mathsf {J}^{n^{\prime },\pm }_r(\mathrm {d}\nu ,\mathrm {d}x)\, \mathrm {d}r\)  converges to \(\mathsf {J}^{\pm }_r(\mathrm {d}\nu ,\mathrm {d}x)\, \mathrm {d}r\)  in \(\mathcal {M}_{loc}(\Gamma \times \mathcal {T}\times [s,t])\) , and
\( \int _0^T \int _{\Gamma \times \mathcal {T}} \tilde{\phi }\left(\frac{1}{3M} \int _{\Gamma \times \mathcal {T}} (1+\nu (X)^2)^{-1}\,\mathsf {J}^{\pm }_t(\mathrm {d}\nu ,\mathrm {d}x) \right) \, \mathrm {d}t < \infty . \) 
Similarly, since the metric \(W\)  is narrowly lower semicontinuous and induces narrow convergence on narrowly pre-compact subsets, we find by an Arzela-Ascoli argument and the estimate (REF ) that, up to choosing a subsequence \(n^{\prime \prime }\) , there exist a narrowly continuous curve \((\mathsf {P}_t)_{t\in [0,T]}\)  such that \(\mathsf {P}^{n^{\prime \prime }}_t\)  converges to \(\mathsf {P}_t\)  for all \(t\in [0,T]\) .
All that remains is showing that \((\mathsf {P},\mathsf {J}^+,\mathsf {J}^-)\in \mathsf {CE}_{\infty }\) . Therefore, fix any \(s,t\in [0,T]\)  and \(F \in \mathrm {Cyl}_c(\Gamma )\) . It is straightforward to verify that there exist constants \(K_{F}\)  and \(C_F\)  such that the following Taylor approximation holds:
\( \left| \mathrm {grad}_{\Gamma }(\nu ,x)\mp n\left(F(\nu {\pm }\genfrac{}{}{}2{1}{n}\delta _x)-F(\nu )\right)\right|\le \genfrac{}{}{}2{C_{F}}{n} 1_{\nu (\mathcal {T})\le K_F}(\nu ,x), \qquad \mbox{ for all }\nu \in \Gamma ,\, x\in \mathcal {T}. \) 
Thus, we can take the limit in the continuity equation \(\mathsf {CE}_n\) , to conclude that
\(\int _{\Gamma } F(\nu ) \,\mathrm {d}\mathsf {P}_t - \int _{\Gamma } F(\nu ) \,\mathrm {d}\mathsf {P}_s &=\lim _{n\rightarrow \infty }\int _{\Gamma } F(\nu ) \,\mathrm {d}\mathsf {P}^{n^{\prime \prime }}_t - \int _{\Gamma } F(\nu ) \,\mathrm {d}\mathsf {P}^{n^{\prime \prime }}_s \\&= \lim _{n\rightarrow \infty } \int _s^t \left( \int _{\Gamma \times \mathcal {T}} (\overline{\nabla }^{n^{\prime \prime },+}F)\,\mathrm {d}\mathsf {J}_r^{n^{\prime \prime },+}+(\overline{\nabla }^{n^{\prime \prime },-}F)\, \mathrm {d}\mathsf {J}_r^{n^{\prime \prime },-} \right) \, \mathrm {d}r \\&=\int _s^t \Big (\int _{\Gamma \times \mathcal {T}} (\mathrm {grad}_{\Gamma } F)\,\mathrm {d}\mathsf {J}_r^+-(\mathrm {grad}_{\Gamma } F)\, \mathrm {d}\mathsf {J}_r^- \Big ) \, \mathrm {d}r,\) 

thereby concluding the proof.
